2016 Spring: The Final Exam of Digital Communications
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1 2016 Spring: The Final Exam of Digital Communications The total number of points is Image of Transmitter Transmitter L 1 θ v 1 As shown in the figure above, a car is receiving a signal from a remote transmitter at distance L 1. The car is moving at speed v parallel to a series of high mountains. The transmitter is located between the mountain line and car directional line. Thus a two-path fading channel is formed. Assume that the length of the reflection path, which is reflected with respect to the high mountain line, is L 2. Let the attenuation of the direct path be α 1 (t and that of the reflection path α 2 (t. Denote the the path delay of the direct path by τ 1 (t and that of the reflection path τ 2 (t. Also denote the angle between the car directional line and the incoming signal from the reflection path by θ 2. (a (6 pt. By setting that τ 1 (0 = L 1 /c and f m v /c, where c is the light speed and is the carrier frequency of the signal wave, prove that Note: Similarly, with τ 2 (0 = L 2 /c, we have τ 1 (t τ 1(0 t + τ 1 (0 = f m cos(θ 1 t + f m L 1 v. (1 τ 2 (t τ 2(0 t + τ 2 (0 = f m cos(θ 2 t + f m L 2 v. (2 1
2 (b (6 pt. The passband channel impulse response of the two-path fading channel is given by c(τ; t = α 1 (t δ (τ τ 1 (t + α 2 (t δ (τ τ 2 (t. Its lowpass equivalent channel impulse response is c l (τ; t = c(τ; te ı 2πfcτ = α 1 (t δ(τ τ 1 (te ı 2πfcτ + α 2 (t δ(τ τ 2 (te ı 2πfcτ = α 1 (t δ(τ τ 1 (te ı 2πfcτ 1(t + α 2 (t δ(τ τ 2 (te ı 2πfcτ 2(t For the path delays, since the time we consider meets t minl 1, L 2 }/c, we can well approximate τ 1 (t T 1 fm L 1 v = L 1 c and τ 2 (t T 2 fm L 2 v = L 2 c. However, for the phases, since is often very large, we cannot ignore the first order terms in (1 and (2 and hence L 1 τ 1 (t f m cos(θ 1 t + f m v = f m cos(θ 1 t + T 1 and L 2 τ 2 (t f m cos(θ 2 t + f m v = f m cos(θ 2 t + T 2. This reduces c l (τ; t to c l (τ; t α 1 (te ı 2πfm cos(θ 1t e ı 2πfcT 1 δ(τ T 1 + α 2 (te ı 2πfm cos(θ 2t e ı 2πfcT 2 δ(τ T 2. Determine R cl ( τ, τ; t + t, t = E c l ( τ; t + tc l (τ; t} under the assumption that α 1 (te ı 2πfm cos(θ 1t, t R} and α 2 (te ı 2πfm cos(θ 2t, t R} are independent of each other and are both zero-mean wide-sense stationary (WSS processes with autocorrelation functions ζ k ( t = E [ α k (t + te ı 2πfm cos(θ k(t+ t α k (te ı 2πfm cos(θ kt ] Note:, T 1 and T 2 are constants. = E [ α k (t + tα k (te ı 2πfm cos(θ k( t ] k = 1, 2. (c (3 pt. Justify that this c l (τ; t is an uncorrelated scattering channel. (d (3 pt. What is the delay spread of the channel (e (4 pt. Determine the spaced-frequency, spaced-time correlation function of the channel. (f (4 pt. Determine the Doppler power spectrum of the channel, provided that for k = 1, 2, Γ k (λ = ζ k ( te ı 2πλ( t d( t. (g (4 pt. Determine the scattering function of the channel. (h (6 pt. Determine the frequency of fade minimum, i.e., f 0 (t = arg min f R C l (f; t. What is the minimum difference between two fade-minimum frequencies Hint: For nonnegative real numbers α 1 and α 2, α1 e ı β 1 + α 2 e ı β ( = α 2 2 α 1 e ı (β 1 β α 2 = α α2 1 α α 1 cos(β 1 β 2, α 2
3 where the minimum occurs when cos(β 1 β 2 = 1. Hint: C l (f; t = c l (τ; te ı 2πfτ dτ = α 1 (te ı 2πfm cos(θ 1t e ı 2π(f+fcT 1 + α 2 (te ı 2πfm cos(θ 2t e ı 2π(f+fcT 2. (i (3 pt. With L 2 L 1 = 3 km, we learn that T 2 T 1 = 10 5 seconds because c = meter/second. Can we avoid the fade minimum (or the occurrence of deep fade by carefully selecting the carrier frequency of the transmission signal if the required transmission bandwidth is 20 MHz. Justify your answer. 2. Suppose we have a two-path channel model with time-invariant impulse response: c l (τ = α 1 e ı 2πfcT 1 δ(τ T 1 + α 2 e ı 2πfcT 2 δ(τ T 2, where α 1 and α 2 are random variables having certain joint distribution, and T 1 and T 2 are constant path delays with T 2 > T 1. The received signal is thus given by r l (t = c l (τs m,l (t τdτ + z l (t, where z l (t is zero-mean white Gaussian noise with two-sided PSD of height 2N 0. Let 2Eg(t, m = 1; s m,l (t = + 2Eg(t, m = 2 be an antipodal modulated signal, where g(t = 1 T, 0 t < T ; 0, otherwise and 0 < T < T 2 T 1. Assume equal prior probability and that the receiver knows the constant path delays T 1 and T 2. Also assume that the receiver can perfectly estimate α 1 and α 2. (a (4 pt. Draw the tapped delay line model of this channel based on the below structure. Delay T 1 Delay T 2 T 1 Σ r l (t 3
4 Hint: You shall replace each of the seven question marks by a proper term. (b (4 pt. Show that r l (t = ± 2E g(t + z l (t, where g(t = α 1 e ı 2πfcT 1 g(t T 1 + α 2 e ı 2πfcT 2 g(t T 2. (c (4 pt. When doing signal demodulation (See for example Slide 4-5, what should be the basis φ(t chosen for vectorization Note that when doing vectorization, the range of integration is set to be from to. Hint: g(t 2 = g(t g (tdt. (d (6 pt. Determine the ML decision rule for detection (after the vectorization. Hint: The decision rule should be a function of r = r l(tφ (tdt, where φ(t is the basis chosen in (c. Note that r is in general a complex number. You may need to consider whether the real or complex part of r is necessary in the ML decision rule. (e (8 pt. Can we realize the ML decision rule in (d by the Rake receiver If positive, draw the realization diagram. If negative, justify why. Hint: Check whether you can formulate the ML decision rule as the form on Slide (f (8 pt. Determine the error probability of the ML decision rule in (d if α 1 and α 2 are independent random variables having common distribution as Pr[α 1 = 0] = Pr[α 1 = 1] = 1 2. ( d Hint: For binary transmission, recall that Pr(error = Q σ 2 3. (a (6 pt. Show that the product of two independent complex random processes p PN (t and c(t is cyclostatinary if p PN (t is zero-mean wide-sense stationary and c(t is cyclostatinary. (b (6 pt. Show that the time-averaged power spectrum density of the product process is given by S p c (f = S p (u S c (f udu. (c (4 pt. Show that if p PN (t is a white process, so is the product process p PN (tc(t, provided that S c (fdf is finite. Hint: Use (b. 4. (a (4 pt. Suppose a direct sequence spread spectrum (DSSS system employs the (n 1, k = (7, 4 Hamming code as the outer code and (n 2, 1 repetition code as the inner code. Select a processing gain L c, which lies between 100 and 110 and which can result in an integer n 2 that fits the DSSS system. (b (4 pt. The 16 codewords of the (7, 4 Hamming code are listed below. 4
5 Message Codeword Message Codeword Derive the coding gain of the DSSS system. (c (6 pt. Given that γ b = 9.6 db satisfies Q( 2γ b = 10 5, find the jamming margin to achieve error rate for this DSSS system via the union bound formula: ( Pr(error (M 1Q where M is the number of (outer codewords. Hint: Follow Slide and check ( Pr(error (M 1Q L c L c 4 min J av /P R cw m av 2 m M 4 min J av /P R cw m av 2 m M Hint: 10 log 10 (2 = 3.01 db and 10 log 10 (12/7 = 2.34 db., L c log 10 (L c db (a (6 pt. Denote a set of Q waveforms as κe ı 2π k U t : t [0, T, } k = 0, 1,..., Q 1 where T is the symbol duration, κ = 1/ T and U is a waveform parameter satisfying U > T. Are these waveforms orthogonal signals Justify your answer. Hint: The inner product of two signals are defined as f(t, g(t = T 0 f(tg (tdt. (b (6 pt. Form the transmission signal via a linear combination of the Q signal waveforms in (a as follows: κ Q 1 s l (t = X ke ı 2π k U t, t [0, T 0, otherwise and transmit it via a noiseless link. Let the received signal be r l (t = s l (t c l (t = c l (τ s l (t τdτ, where s l (t is the periodic counterpart of s l (t with period T. Show that Q 1 ( r l (t = κ X k e ı 2π k k U t C l, U where C l (f = c l(τe ı2πfτ dτ. 5
6 (c (8 pt. Sampling r l (t with sampling period T/N, where N > Q, we obtain ( m Q 1 r m = r l N T = κ C l ( k U We then perform N-point DFT onto r m } N 1 m=0, i.e., R n = kmt ı 2π X k e UN m = 0, 1,..., N 1 N 1 m=0 mn ı 2π r m e N. Can we recover each of X k } Q 1 from R n} N 1 n=0 for every T and U satisfying U > T if C l (k/u} Q 1 are known Justify your answer. (d (8 pt. Re-do (c by sampling r l (t with sampling period U/N, where N > Q. Note that in such case, ( m Q 1 r m = r l N U = κ C l ( k U X k e ı 2π km N m = 0, 1,..., N 1. 6
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